It is the second hurricane of the June-to-November Atlantic hurricane season, which is predicted to be especially active this year. Some 23,000 people from coastal areas in the Mexican state of Tamaulipas have already left their homes before Dolly's destructive force hits.
Meanwhile, some oil drilling companies have evacuated workers from their offshore rigs - causing world oil prices to creep up slightly.
"The first storm in always gets the adrenaline pumping, and it helps bring everybody into the mind set for hurricane season," BP spokesman Tom Mueller told the Houston Chronicle.
A Category One storm is the lowest rating in the five-level Saffir-Simpson scale, but the National Hurricane Centre in Florida still predicted six to 10 inches (15-25cms) of rain over south Texas and north east Mexico in the coming days.
